Summary of part 1

The two bishops and the attacking king can mate the defending king but only in the corner.

Whilst other theoretical mates are available, it's not sensible to expect your opponent to "walk" into them.

Part 2 of 3 - Pushing the king to the corner, creating fences

1...Bd6 is a lovely move which deprives the white king of escape squares and forces him to the 8th rank. This will definitely help us in our quest.

1...Bg3 is another helpful idea. We have two fences running from h2-b8 and h1-a8. This forces the white king to the back rank and closer to his eventual fate!

Now, let's see if we can force a mate from this last position.

[Event "Study of Two Bishops and King v King"] [Site "Chess Toolkit (Author = TQM.)"] [Date "2017.02.19"] [Round ""] [White "A King"]
[Black "T Bishops"] [Result "0-1"] [ECO "N/A"] [Annotator "TQM"] [SetUp "1"]
[FEN "2K5/8/4k3/8/4b3/8/8/4b3 b"] [PlyCount "44"] [EventDate "2017.02.19"] [EventType "Endgame Study"] [EventCountry "GB"]
1...Bg3 {our solution above} 2.Kd8 Bb7 {creating another fence a6-c8} 3. Ke8 Bc7 {and another from a5-d8. The king is frogmarched towards the corner} 4. Kf8 Kf6 5. Ke8 {of course the defender will kick and scream to avoid being mated} 5...Bc6! {unlucky, defender! As Bc7 covers d8, the only way is to the corner. The king covers lots of nice squares now so it's all very mechanical!} 6. Kf8 Bd6 7. Kg8 Kg6! {now white can only alternate between g8 and h8} 8. Kh8 {what to do now! Well, let's say the king was on g8 instead, we could play ...Bd5+ and ...Be5# but it's not. However, we have so many squares on the diagonals where we can move the bishops to waste a move - this would then put the king on g8 and we know the plan!}8...Bb4 (8...Bd5??{Stalemate}){any bishop move on the a3-e7 diagonal is fine, as is any sensible light square bishop move} 9. Kg8 {where we want him!} Bd5! 10. Kh8 Bc3

In the final section, we will look at how to push the king to the side before exercising this technique!

Summary of part 2

We have learned some techniques to help push the king back.

Creating a "fence" or border to restrict the king is very helpful.

Sometimes we will need to be patient and waste a move to avoid stalemating!

We know how to mate the king from the side. Now we need to know how to get the king to the side of the board.

Part 3 of 3 - Winning Technique

[Event "Study of Two Bishops and King v King"] [Site "Chess Toolkit (Author = TQM.)"] [Date "2017.02.19"] [Round ""] [White "A King"]
[Black "T Bishops"] [Result "0-1"] [ECO "N/A"] [Annotator "TQM"] [SetUp "1"]
[FEN "8/8/8/8/5K2/7b/1k6/b7 b"] [PlyCount "44"] [EventDate "2017.02.19"] [EventType "Endgame Study"] [EventCountry "GB"]
1...Kc2 {the position is far from ideal for the attacker - all his pieces are attacking random diagonals. The first thing we'll do is restrict the enemy king as much as we can with our own. We move to c2 rather than c3 to allow the bishop to control the long diagonal} 2. Ke3 (1...Kg3 {just puts the king closer to the corner and will not help the defender})2...Bf5 {the bishops now work in harmony to cut the board diagonally in half - the king can no longer enter the board to the left of the b1-h7 diagonal.}
3. Kf4 Bd3 {keeping the two diagonals closed} 4. Ke3 {trying to hold the attacker back} 4...Bb2! {super - passing the move to the defender who was well placed} 5. Kf3 Bc1 {now we have a smaller border - the king cannot go left of the c1-h6 diagonal!} 6. Kg4 Kd1 {we bring the king closer without disrupting the scope of our bishops} 7. Kf3 Bd2 8. Kg4 Ke2 9. Kg3 Bf5! {forcing the king to the side of the board or closer to the corner!} 10. Kg2 (10. Kh4 {we execute our technique from section 2} Kf3 11. Kh5 Bc2 12. Kh4 Bg6 13. Kh3 Bg5 14. Kh2 Kf2 15. Kh3 Bf5 16. Kh2 Bf4 17. Kh1 Be4) 10...Be1! 11. Kh2 Kf2 {now we are in the ideal place with the king just oscillating between h2 and h1!} 12. Kh1 Bc3 13. Kh2 Be5 14. Kh1 Be4

Note lines given as instructive, rather than optimal engine choices.

Summary

This is a won ending

The technique is to push the king into one corner, using the bishops as a fence

Once the king is on the side of the board, we can gently push him to the corner by force.

We need the king and both bishops to help mate the king at the side of the board.

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